Question

When two fluid streams are mixed in a mixing chamber, can the mixture temperature be lower than the temperature of both streams? Explain.

Short answer

Answer: No, the temperature of the mixture will lie between the initial temperatures of the two fluids due to the conservation of energy and heat exchange during the mixing process. It cannot be lower than the temperature of both initial streams.

Step-by-step solution

Understand the problem

We are asked to determine whether the mixture's temperature can be lower than both streams' temperature when two fluid streams are mixed in a mixing chamber. To reach a conclusion, we must consider the thermodynamics of the mixing process.

Energy conservation during mixing

In a closed system, energy conservation is applied, which means the total internal energy of the system remains constant throughout the mixing process. Therefore, the energy released by the hotter fluid must be equal to the energy gained by the colder fluid.

Heat exchange between the fluids

When two fluids with different temperatures mix, heat exchange occurs between the fluids until they reach thermal equilibrium. This means that the hotter fluid will lose heat, and its temperature will decrease while the colder fluid will gain heat, and its temperature will increase. Given that the energy lost by the hotter fluid equals the energy gained by the colder fluid, the mixture's final temperature will lie between the initial temperatures of the two fluids. That is, it will be higher than that of the colder fluid and lower than that of the hotter fluid.

Conclusion

When two fluid streams with different temperatures mix in a mixing chamber, the mixture's temperature cannot be lower than the temperature of both streams. It will be in between the initial temperatures of the two fluids due to the conservation of energy and heat exchange during the mixing process.

Key concepts

Energy Conservation in Thermodynamic Processes

The principle of energy conservation is foundational to understanding thermodynamics, especially when it comes to analyzing the outcome of mixing different temperature fluids. It states that energy cannot be created or destroyed, only transferred or converted from one form to another. During a mixing process, the total internal energy within the closed system must stay constant.

Take, for example, the scenario where two fluid streams at distinct temperatures are mixed in a chamber. The warmer stream will release some of its thermal energy, while the colder stream absorbs it. This energy ‘exchange’ obeys the law of conservation; the energy given up by the hotter fluid is accounted for as the gain for the cooler fluid, without any loss or addition of energy within the system. If we were to measure the energy before and after mixing, the total would be equal, highlighting this bedrock principle's pivotal role in leading us to the conclusion that the mix cannot be cooler than both original stream temperatures.

Thermal Equilibrium in Mixing

When discussing thermal equilibrium, the focus is on the end-state of a thermodynamic process. In the context of mixing two different fluids, thermal equilibrium is the point at which both fluids have exchanged heat to the extent that their temperatures even out and become the same.

It's akin to reaching a peaceful compromise in a negotiation; the initially hotter fluid cools down as it loses thermal energy, while the initially cooler fluid warms up by gaining that energy. This mutual exchange continues until no further temperature changes occur, and once this state is achieved, the fluids are in thermal equilibrium. Consequently, because no more heat is flowing between the fluids, their common temperature must naturally fall between the two starting temperatures—illustrating why the mixed temperature cannot be lower than the initial temperature of both separate streams.

Heat Exchange Mechanisms

The heat exchange is the mechanism by which thermal energy is transferred between substances or objects at different temperatures. When fluids of dissimilar temperatures come into contact within a mixing chamber, heat naturally flows from the hotter fluid to the cooler one.

This process continues until the thermal energy distribution reaches a balance—thermal equilibrium. The manner and rate of this heat transfer can vary, influenced by factors such as the temperature difference, the specific heat capacities of the fluids involved, and the volume flow rates.

Understanding this concept helps shed light on why, in our example, the resulting mixture's temperature cannot be lower than the lower temperature of the initial streams. Heat naturally flows from hot to cold, not the reverse, and thus it's impossible to end with a temperature that's below that of the initially cooler fluid, given the mix occurs within an isolated system with no additional energy sinks or sources.